m7 solid electrolytes l1
DESCRIPTION
M7 Solid Electrolytes L1TRANSCRIPT
L2: Solid Electrolytes
Professor S.R. Elliott and Dr S.N. Taraskin
Lecture 1: Structure of Disordered Materials
Lecture Synopsis• Lecture 1: Structure of Disordered Materials (SRE)
• Lecture 2: Defects (SRE)
• Lecture 3: Examples of Solid Electrolytes (SRE)
• Lecture 4: Mechanisms of Diffusion (SNT)
• Lecture 5: Fundamentals of Diffusion (SNT)
• Lecture 6,7: Random Walk and Approximations (SNT)
• Lecture 8: Conductivity and Disorder (SNT)
• Lecture 9: Depolarization Current (SNT)
• Lecture 10: Impedance Spectroscopy and Radiotracers (SRE)
• Lecture 11: NMR and QENS (SRE)
• Lecture 12: Applications (SRE)
• Solid electrolytes - or superionic conductors - or fast-ion conductors - are materials in which the ionic (cationic/anionic) conductivity is comparable to that of liquid electrolytes, and the electronic conductivity is negligible
• In solids, atomic diffusion can only occur via structural defects - so need to understand disorder
• Disorder is defined with respect to a reference structure - eg the ideal crystal
Salvador Dali (1952)
“Galateaof theSpheres”
The Physics & Chemistry of Solids
SRE
Crystals
Ruavb wc
•An ideal crystal is generated by the translationally periodic repeat of a unit cell.•A lattice is an infinite array of mathematical points having the translational periodicity of the crystal•The unit cell is defined by the vectors a, b, c (in 3D)•Any 2 lattice points are connected by the vector
•Unit cells can be: primitive (P), body-centred (I), side-centred (C), face-centred (F)
Disordered Materials• Two classes of materials have topologically disordered structures:
liquids & glasses
Liquids• Topological disorder plus dynamical disorder (orientational and
translational)
Topological Spin
Substitutional Positional/vibrational
Types of disorder
• Slow cooling of a melt -> crystallization- crystal nuclei grow into stable crystallites in liquid, before the viscosity (exponentially) increases with decreasing temperature
• Crystallization of melt is a first-order thermodynamic phase transition (density discontinuity)
Crystallization
Glasses/Amorphous Solids• Topological (quenched) disorder
• Rapid cooling of a melt supercooled liquid- rapid increase with viscosity with decreasing temperature
precludes crystal nucleation and growth transforms to a glass at Tg (glass-transition temperature)
glass = a “solid” liquid on the experimental timescale amorphous = non-crystalline
Vitrification
Structure of Liquids and Glasses• Structure has no long-range order (LRO): no lattice, no unit cell.• But structure is (generally) orientationally isotropic • Statistical description in terms of spatial correlation functions- e.g. pair distribution functions, i.e. the atomic density function, ρ(r)
- Radial distribution function (RDF) is defined as:
J r 4r 2 r -it is the average probability of finding an atom in the distance interval r → r + dr from a given atom at r = 0
• Area under an RDF peak gives the average atomic coordination number• Positions of RDF peaks give radii of neighbour shells
Short-range Order (SRO)• Absence of LRO in glasses and liquids does not mean that there
is no structural order whatsoever• Glasses and liquids can have (varying degrees of) SRO
- e.g. corner (O)-shared SiO4 tetrahedra in g-SiO2
- well-defined coordination numbers, bond lengths, bond angles
• Bond-length and bond-angle disorder (c.f. widths of first 2 RDF peaks) are not generally sufficient alone to destroy LRO
• Instead, fluctuations in the dihedral angle, φ, destroy correlations between nearest-neighbour structural units for covalent systems.
P( )
ideal amorphous
ideal crystal
Dihedral angle
Structure Determination of Glasses and Liquids
n 2dhkl sin
•X-ray, neutron or electron diffraction (scattering plus interference) is commonly used to study the atomic structure of materials•For the case of crystals, with well-defined lattice planes (hkl), Bragg diffraction occurs:
• A crystal acts like a 3D diffraction grating for X-rays etc, and diffracted beams only lie in particular directions, at angles θhkl
• For amorphous materials, diffraction occurs over a range of angles- Bragg spots (for crystals) become diffuse haloes (for glasses)
hkl
• Complex amplitude of a wave, scattered by atom i, is:
Fi = fi exp [i(k − k0). ri] ,
where
K k k0 4 sin
• The atomic scattering factor, fi, is constant for neutrons - strongly decreasing function of K or θ for X-rays & e-
wavevector transfer
Nb k0 is incident wavevector k is scattered wavevector
• Total scattering intensity from a collection of (pairs of) atoms (i,j) is generally:
i fi fj exp i k k0 .rij
j
where
rij ri rj
• Amorphous materials are spatially isotropic overall. The orientational average of the phase factor is:
exp i k k0 .rij 1
4rij2 exp iKrij cos
0
2rij2 sind
sin Krij
Krij
,
giving the Debye scattering equation:
I i fi fj
sin KrijKrijj
• For a monatomic solid (fi = fj = f):
I f 2
i
i f 2 sin Krij
Krijj
• Introducing the atomic-density distribution function ρi(rij) (for origin atom i) gives:
I f 2
i f 2 i rij sin Krij
Kriji dVi
where ∑j→∫.• Writing ρi(r) = <ρi(rij)>, and adding and subtracting a term in the macroscopic average atomic density, ρo, gives:
I f 2
i f 2 4r 2 r o sin Kr
Kri dr f 2 4r 2o sin Kr
Kri dr
where ∑i→N.
• The last term in ρo represents scattering due to the finite sample size (coinciding with the θ = 0o transmitted beam)
• Hence
I Nf 2 Nf 2 4r 2 r o sin KrKr
dr
• Defining the structure factor, S(K) as:
S K I K / Nf 2
and the reduced scattering function, F(K), as:
F(K ) K[S(K ) 1]then
F(K ) G(r )sin Kr drwhere the reduced RDF, G(r), is given by:
G(r ) 4r r o J(r )/ r 4ro
Nb Fourier transform
• F(K) and G(r) are therefore related via a Fourier transform
• Can obtain the RDF from measured scattering data I(K) → S(K) → F(K) by an inverse Fourier transform:
KKrKFrG dsin)(2)(
Nb limits: 0 < K < ∞